Modeling Heart Disease Mortality in Space and Time

In Waco and Surrounding Counties

Dusty Turner

Motivating Question

What factors impact heart disease mortality in the United States?

Space

Time

Age

Gender

Other Research

According to the CDC:

Heart Disease has been the leading cause of death from 2009-20191

  1. There has been a 21.3/100,000 person drop in heart disease fatalities (182.8 to 161.5) over the last decade.
  2. Males were nearly twice as likely as females to die of heart disease.
  3. Heart disease death rates are highest among Black, non-hispanics and lowest among Asian and Pacific Islanders.

For perspective, during the height of COVID, it accounted for 1 in 8 deaths (or 697,000 deaths) in the United States. Heart disease was the number one cause of death, followed by cancer, with these two causes of death accounting for a total of 2.15 million deaths.2

Other Factors

Other research has shown other factors to impact heart disease fatalities:

  1. Personal diet: Ulbricht and Southgate (1991)
  2. Stress: Bunker et al. (2003)
  3. Sleep quality and duration: Lao et al. (2018)
  4. Hereditary factors: Bates et al. (2003)
  5. Culture: Syme, Hyman, and Enterline (1964)

Explore the data

Heart Disease Deaths Over Time

Heart Disease Deaths By Gender and Age

Heart Disease Deaths Compared to Ozone Coverage

Mean estimated 8-hour average ozone concentration in parts per billion (ppb) within 3 meters of the surface of the earth

Models to Explore

  1. Independent Error Model
    – Linear Regression

  2. Spatial Only Models
    – Spatial Model, no nugget effect {bbstdr}
    – Spatial Model, nugget effect {spBayes} and {INLA}

  3. Spatio-Temporal Models
    – Separable spatio-temporal model without any nugget effect
    – Spatio-temporal model fitting with {spTimer}
    – Autoregressive model fitting with {INLA}
    – Spatio-temporal dynamic models using {spTDyn}

Fitting independent error regression models

\(y|x ∼ N(\beta_0 + \beta_i(x), \sigma^2)\)

\(\beta_i \sim N(0,10,000)\)

\(\sigma^2 \sim Gamma(\alpha = 2,\beta = 1)\)

Params mean sd 2.5% 97.5% Contains_0
(Intercept) 372.2 321.6 −258.6 1,002.9
stratification1Ages 65 years and older 1,422.0 25.1 1,372.7 1,471.3 *
stratification3Women −292.4 25.1 −341.7 −243.1 *
ozone −2.1 7.6 −16.9 12.8
sigma2 48,649.5 3,920.3 41,566.4 56,919.1 *

Spatial Model; no Nugget Effect

\(Y \sim N_n(X\beta,\sigma^2_ϵH)\)

\(H_{ij} = exp(−\phi d_{ij})\)

\(d_{ij}\) is the distance between location \(s_i\) and \(s_j\).

Params mean sd 2.5% 97.5% Contains_0
(Intercept) 326.8 325.5 −311.5 965.1
stratification1Ages 65 years and older 1,418.3 25.6 1,368.1 1,468.5 *
stratification3Women −307.7 26.3 −359.3 −256.0 *
ozone −0.7 7.7 −15.7 14.4
sigma2 58,496.6 4,713.8 49,979.8 68,440.1 *

Fitting spatial models with nugget effect

With {spBayes}

\(Y(s_i)=x'(s_i)\beta+w(s_i)+\epsilon(s_i)\)

Nugget Effect: \(\epsilon(s_i) \sim N(0,σ^2_ϵ)\)

Params mean sd 2.5% 97.5% Contains_0
X.Intercept. 35.0 96.0 −149.6 228.8
stratification1Ages.65.years.and.older 1,334.5 25.8 1,283.6 1,384.1 *
stratification3Women −272.7 24.6 −320.1 −222.5 *
ozone 6.7 2.3 2.0 11.3 *
sigma.sq 0.9 1.5 0.2 4.0 *
tau.sq 51,091.8 4,397.8 43,122.4 60,622.5 *
phi 0.9 0.5 0.1 1.9 *

Fitting spatial models with nugget effect

With {INLA}

Params mean sd 2.5% 97.5% Contains_0
X.Intercept. 2.0 31.7 −60.1 64.5
stratification1Ages.65.years.and.older 214.9 30.7 152.6 273.9 *
stratification3Women −44.7 29.2 −101.0 14.1
ozone 17.9 1.3 15.5 20.4 *
phi 1.0 0.4 0.4 1.9 *
sigmasq 0.3 0.4 0.0 1.3 *
tausq 458,895.2 34,270.6 386,795.9 520,936.6 *

Compare Models Through Cross Validation

Model Scoring

model rmse mae cvg Comp_Time
spBayes 256.41 205.59 90.71 4.00
none 296.04 241.71 82.14 8.34
none 308.36 255.93 83.57 13.72
inla 470.48 422.22 100.00 20.42

Spatio Temporal

Seperable Spatio-Temporal Model Specification

\(\mu_{it} = \alpha + \epsilon_i + S_i + \gamma_t + \tau_t\)

\(\mu_{it}\): Mean response at location \(i\) at time \(t\)
\(\epsilon_i \sim N(0,\sigma^2\epsilon)\): Unstructured spatial effect
\((S_1, .., S_l)\): Spatial Model
\(\gamma_t \sim N(0, \sigma^2_\gamma)\): Unstructured temporal effect
\((\tau_1,...,\tau_T)\): time series model

\(\phi_t\): Temporal decay parameter: 3
\(\phi_s\): Spatial decay parameter: 0.0121

Seperable Spatio-Temporal Model

Params mean sd 2.5% 97.5% Contains_0
(Intercept) 142.5 548.5 −933.3 1,218.3
stratification1Ages 65 years and older 1,437.8 23.5 1,391.8 1,483.9 *
stratification3Women −294.1 26.2 −345.5 −242.6 *
ozone 1.8 7.4 −12.7 16.3
sigma2 753,893.3 60,750.5 644,129.8 882,042.6 *

Inseperable Spatio-temporal Model

re write this to show 3.11 from this link

https://www.sujitsahu.com/bmbook/bmstdr-full_vignette.html#modeling-temporal-areal-unit-data

\(\mu_{it} = \alpha + \epsilon_i + S_i + \gamma_t + \tau_t + \delta_{it}\)

\(f(\delta|\sigma^2_{\delta}) \propto -\frac{1}{2\sigma^2_{\gamma}}\Sigma_{t=2}^T\Sigma_{i\sim j}(\delta_{ij}-\delta_{jt}-\delta_{i,t-1}+\delta_{j,t-1})^2\)

Inseperable Spatio-temporal Model {spTimer}

sig2eps: random effect error
sig2eta: spatial error
phi: spatial decay parameter

Params mean sd 2.5% 97.5% Contains_0
(Intercept) 29.7 94.3 −150.3 207.3
stratification1Ages 65 years and older 1,336.1 25.1 1,288.5 1,384.5 *
stratification3Women −274.6 24.8 −323.8 −227.1 *
ozone 6.8 2.3 2.4 10.9 *
sig2eps 0.0 0.0 0.0 0.0 *
sig2eta 51,543.4 4,403.9 43,570.2 60,959.3 *
phi 1.7 1.3 0.5 5.0 *

Inseperable Spatio-temporal Model {INLA}

Params mean sd 2.5% 97.5% Contains_0
stratification1Ages.65.years.and.older 206.4 30.1 146.2 265.0 *
stratification3Women −41.9 28.0 −96.2 14.6
ozone 18.0 1.1 16.0 20.2 *
sigma2eps 429,388.9 34,289.1 361,431.0 496,460.9 *
sig2eta 0.0 0.2 0.0 0.4 *
phi 9.4 6.1 1.7 23.8 *

Inseperable Spatio-temporal Model {spTDyn}

Params mean sd 2.5% 97.5% Contains_0
(Intercept) −62.8 97.7 −250.5 121.4
stratification1Ages 65 years and older 838.5 37.5 764.8 913.1 *
stratification3Women −177.8 37.2 −249.9 −106.4 *
ozone 11.6 2.5 6.6 16.4 *
sig2eps 209,723.3 31,518.8 152,556.5 278,185.8 *
sig2eta 176,242.5 12,585.7 154,004.8 203,732.4 *
phi 0.9 0.1 0.7 1.0 *

Compare Models Through Cross Validation

Model Scoring

model rmse mae cvg Comp_Time
spTimer 190.10 169.27 100.00 44.00
spTDyn 222.85 195.10 100.00 32.00
inla 475.53 426.94 12.86 10.32

Bibliography

STA 5377: Spatial Statistics

Bates, Benjamin R., Alan Templeton, Paul J. Achter, Tina M. Harris, and Celeste M. Condit. 2003. “What Does ‘a Gene for Heart Disease’ Mean? A Focus Group Study of Public Understandings of Genetic Risk Factors.” American Journal of Medical Genetics Part A 119A (2): 156–61. https://doi.org/https://doi.org/10.1002/ajmg.a.20113.
Bunker, Stephen J, David M Colquhoun, Murray D Esler, Ian B Hickie, David Hunt, V Michael Jelinek, Brian F Oldenburg, et al. 2003. “‘Stress’ and Coronary Heart Disease: Psychosocial Risk Factors.” Medical Journal of Australia 178 (6): 272–76. https://doi.org/https://doi.org/10.5694/j.1326-5377.2003.tb05193.x.
Lao, Xiang Qian, Xudong Liu, Han-Bing Deng, Ta-Chien Chan, Kin Fai Ho, Feng Wang, Roel Vermeulen, et al. 2018. “Sleep Quality, Sleep Duration, and the Risk of Coronary Heart Disease: A Prospective Cohort Study with 60,586 Adults.” Journal of Clinical Sleep Medicine 14 (01): 109–17. https://doi.org/10.5664/jcsm.6894.
Syme, S.Leonard, Merton M. Hyman, and Philip E. Enterline. 1964. “Some Social and Cultural Factors Associated with the Occurrence of Coronary Heart Disease.” Journal of Chronic Diseases 17 (3): 277–89. https://doi.org/https://doi.org/10.1016/0021-9681(64)90155-9.
Ulbricht, T. L. V., and D. A. T. Southgate. 1991. “Coronary Heart Disease: Seven Dietary Factors.” The Lancet 338 (8773): 985–92. https://doi.org/https://doi.org/10.1016/0140-6736(91)91846-M.